1
$\begingroup$

So, this is a specific task that states:

We're given a rectangular conductive contour with known dimensions $a=5cm$ and $b=7.5cm$ like in the picture. At the moment, the distance between the contour and a very long wire conductor, with known current $I=2A$, is $c=2.5cm$. The resistance of rectangular contour is $R=2\Omega$. At one point, the contour starts moving to the right (staying in the plane of the picture at the same time) and stops when it moves the distance $b$. What's the total charge that travels through the contour from the beginning moment to when it stops moving?

enter image description here

So, what I did first is stated that the magnetic field around the long wire conductor at some distance $r$ is: $$\oint_C\vec{B}d\vec{l}=\mu_0\sum_C I$$ $$\oint_C B\cdot dl\cdot \cos<(\vec{B},d\vec{l})=\mu_0\sum_C I$$ $$\oint_C Bdl=\mu_0I$$ $$B\cdot 2\pi r=\mu_0 I$$ $$B=\frac{\mu_0 I}{2\pi r}$$ Now the magnetic flux through the surface of the rectangle would be: $$\Phi=\int_S \vec{B}d\vec{S}=\int_S B\cdot dS\cdot cos<(\vec{B},d\vec{S})=\int_S B\cdot dS=$$ $$=\int_{c}^{c+b}\frac{\mu_0 I}{2\pi r}\cdot a\cdot dr=\frac{\mu_0 I a}{2\pi}\ln{\frac{c+b}{c}}$$ Now, what confuses me is that I don't know the exact time interval the contour traveled from the first to second position, nor do I understand if it's static or dynamic induction, or both that occur during the movement. I had the idea of calculating the electromagnetic induction that occurred during the movement, then calculating the current due to induction as: $$I_{ind}=\frac{e_{ind}}{R}$$ and then calculating the total charge that went through the contour as: $$Q=\int_{\Delta t}Idt$$ But for all of that I need the time interval in which the contour moved, or even the speed at which it moved, from which I could calculate the time.
Can this be solved in any other way?

$\endgroup$
0
$\begingroup$

$\mathcal E = (-) \dfrac{d\Phi}{dt}$ and $I= \dfrac{\mathcal E}{R} \Rightarrow I = \dfrac 1 R \dfrac{d\Phi}{dt} = \dfrac {dQ}{dt}$

$\displaystyle\int_0^Q dQ = \dfrac 1 R \int^{\Phi_{\rm final}}_{\Phi_{\rm initial}}d\Phi$

This is a way of measuring changes in magnetic flux and hence magnetic field strength of a uniform field if you have a (search) coil and a charge measuring device like a ballistic galvanometer.
Here is a brief outline of such an arrangement.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.